Summary: Ebola virus disease is a lethal human and primate disease that currently requires a particular attention from the international health authorities due to important outbreaks in some Western African countries and isolated cases in the UK, the USA and Spain. Regarding the emergency of this situation, there is a need for the development of decision tools, such as mathematical models, to assist the authorities to focus their efforts in important factors to eradicate Ebola. In this work, we propose a novel deterministic spatial-temporal model, called Between-Countries Disease Spread (Be-CoDiS), to study the evolution of human diseases within and between countries. The main interesting characteristics of Be-CoDiS are the consideration of the movement of people between countries, the control measure effects and the use of time-dependent coefficients adapted to each country. First, we focus on the mathematical formulation of each component of the model and explain how its parameters and inputs are obtained. Then, in order to validate our approach, we consider two numerical experiments regarding the 2014-2015 Ebola epidemic. The first one studies the ability of the model in predicting the EVD evolution between countries starting from the index cases in Guinea in December 2013. The second one consists of forecasting the evolution of the epidemic by using some recent data. The results obtained with Be-CoDiS are compared to real data and other model outputs found in the literature. Finally, a brief parameter sensitivity analysis is done. A free MATLAB version of Be-CoDiS is available at: http://www.mat.ucm.es/momat/software.htm.